c,數(shù)據(jù)庫(kù)數(shù)學(xué)建模老的不用了chap-04e

1、1Truncation Errors and the Taylor SeriesChapter 42How does a CPU compute the following functions for a specific x value? cos(x) sin(x) ex log(x) etc.Non-elementary functions such as trigonometric, exponential, and others are expressed in an approximate fashion using Taylor series when their values,

2、derivatives, and integrals are computed.Taylor series provides a means to predict the value of a function at one point in terms of the function value and its derivatives at another point.3Define the step size as h=(xi+1- xi), the series es:Taylor Series (nth order approximation):The Reminder term, R

3、n, accounts for all terms from (n+1) to infinity.nth order approximation:Second order approximation: first order approximationAny smooth function can be approximated as a polynomial.Take x = xi+1 Then f(x) f(xi) zero order approximation Each additional term will contribute some improvement to the ap

4、proximation. Only if an infinite number of terms are added will the series yield an exact result.In most cases, only a few terms will result in an approximation that is close enough to the true value for practical purposes5ExampleApproximate the function f(x) = 1.2 - 0.25x - 0.5x2 - 0.15x3 - 0.1x4 f

5、rom xi = 0 with h = 1 and predict f(x) at xi+1 = 1.6Choose x = xi+1 and xi = 0 Then f(xi+1) = f(x) and (xi+1 xi) = xSince First Derivative of ex is also ex :(2.) (ex )” = ex (3.) (ex)” = ex, (nth.) (ex)(n) = exAs a result we get:Example: computing f(x) = ex using Taylor Series expansionLooks familia

6、r?Maclaurin series for ex 7Choose x=xi+1 and xi=0 Then f(xi+1) = f(x) and (xi+1 xi) = xDerivatives of cos(x): (1.) (cos(x) ) = -sin(x) (2.) (cos(x) )” = -cos(x), (3.) (cos(x) )” = sin(x) (4.) (cos(x) )” = cos(x), As a result we get:Yet another example:computing f(x) = cos(x) using Taylor Series expa

7、nsion8Notes on Taylor expansion:Each additional term will contribute some improvement to the approximation. Only if an infinite number of terms are added will the series yield an exact result.In most cases, only a few terms will result in an approximation that is close enough to the true value for p

8、ractical purposesReminder value R represents the truncation errorThe order of truncation error is hn+1 R=O(hn+1), If R=O(h), halving the step size will halve the error.If R=O(h2), halving the step size will quarter the error.9*here Error PropagationLet xfl refer to the floating point representation of the real number x. Since computer has fixed word length, there is a difference between x and xfl (round-off error) and we would like to estimate the error in the calculation of f(x) :Both x and f(x) are unknown. If xfl is close to x, then we can use first order T